A Regularity Criterion for Semigroup Rings

An analogue of the Kunz-Frobenius criterion for the regularity of a local ring in a positive characteristic is established for general commutative semigroup rings. Let S be a commutative semigroup (we always assume that S contains a neutral element), and K a field. For every m 6 Z+ the assignment x H-» x, x £ S, induces a K-endomorphism 7m of the semigroup ring R = K[S]. Therefore we can consider R as an .R-algebra via 7rm, and especially as an R-module. Let R[m] denote R with its R-module structure induced by rm. If S is finitely generated, then R [m] is obviously a finitely generated .R-module. In this note we want to give a regularity criterion for S in terms of the homological properties of R that is analogous to Kunz's [1] characterization of regular local rings of a characteristic p > 0 in terms of the Frobenius functor. Our criterion, which generalizes the result of Gubeladze [2, 10.2], requires only a mild condition on S and we provide a 'pure commutative algebraic' proof. (In [2] the result was stated for seminormal simplicial affine semigroup rings and derived from the main result of [2] that K -regularity implies the regularity for such rings.) Theorem 1. Let S be a finitely generated semigroup, K a field, R= K[S], and m 6 Z+, m > 0. Suppose that S has no invertible element = 1 and is generated by irreducible elements. Then the following conditions are equivalent: (a) R has a finite projective dimension; (b) R is a free module; (c) S is free, in other words, S = Z for some n € Z+. 1991 Mathematics Subject Classification. Primary 13D05, 20M25; Secondary 13A35.